Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T12:00:22.111Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-dimensional bubbles in slow viscous flows

Published online by Cambridge University Press:  28 March 2006

S. Richardson
S. Richardson
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge Present address: Department of Mathematics, University of Manchester Institute of Science and Technology.
Get access Rights & Permissions [Opens in a new window]

Abstract

The representation of a biharmonic function in terms of analytic functions is used to transform a problem of two-dimensional Stokes flow into a boundary-value problem in analytic function theory. The relevant conditions to be satisfied at a free surface, where there is a given surface tension, are derived.

A method for dealing with the difficulties of such a free surface is demonstrated by obtaining solutions for a two-dimensional, in viscid bubble in (a) a shear flow, and (b) a pure straining motion. In both cases the bubble is found to have an elliptical cross-section.

The solutions obtained can be shown to be unique only if certain restrictive assumptions are made, and if these are relaxed the same methods may give further solutions. Experiments on three-dimensional inviscid bubbles (Rumscheidt & Mason 1961; Taylor 1934) demonstrate that angular points appear in the bubble surface, and an analysis is presented to show that such a discontinuity in a two-dimensional free surface is necessarily a genuine cusp and the nature of the flow about such a point is examined.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 33 , Issue 3 , 2 September 1968 , pp. 475 - 493
Copyright
© 1968 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bretherton, F. P. 1962 J. Fluid Mech. 12, 591.
Cherepanov, G. P. 1963a P.M.M. 27, 275.
Cherapanov, G. P. 1963b P.M.M. 27, 428.
Cherepanov, G. P. 1964 P.M.M. 28, 141.
Clarke, N. S. 1966 Ph.D. Thesis, London University.
Clarke, N. S. 1968 J. Fluid Mech. 31, 481.
Fletcher, A. 1940 Phil. Mag. (7), 30, 516.
Garabedian, P. R. 1966 Comm. Pure Appl. Math. 19, 421.
Jahnke, E., Emde, F. & LöSCH, F. 1960 Tables of Higher Functions. New York: McGraw-Hill.
Krakowski, M. & Charnes, A. 1953 Stokes' paradox and biharmonic flows. Carnegie Institute of Technology, Technical Rept. no. 37.Google Scholar
Langlois, W. E. 1964 Slow Viscous Flow. New York: Macmillan.
Michael, D. H. 1958 Mathematika, 5, 82.
Moffatt, H. K. 1964 J. Fluid Mech. 18, 1.
Muskhelishvili, N. I. 1963 Some Basic Problems of the Mathematical Theory of Elasticity. Translated by J. R. M. Radok. P. Noordhoff Ltd.
Proudman, I. & Pearson, J. R. A. 1957 J. Fluid Mech. 2, 237.
Rumscheidt, F. D. & Mason, S. G. 1961 J. Colloid Sci. 16, 238.
Taylor, G. I. 1934 Proc. Roy. Soc. A, 146, 501.
Taylor, G. I. 1964 Proceedings of the 11th International Congress of Applied Mechanics. Editor, H. Görtler. Munich: Springer Verlag.